For hydrogen atoms, all book take correction up to 1/c^2 where the perturbation is -P^4/8*m^3*c^2. And they go solving it by sandwiching p^4 term where they consider p^2 = 2m*(1+e^2/r). and they square it to solve for p^4. To get a better view of perturbation to first order please see attachment. What if I want it to the second order correction, that is to p^6? The additional perturbative term would be p^6/16*m^5*c^4. What should be done in this case?